statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
nhds_coe {r : ℝ} : 𝓝 (r : ereal) = (𝓝 r).map coe | (open_embedding_coe.map_nhds_eq r).symm | lemma | ereal.nhds_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_coe_coe {r p : ℝ} :
𝓝 ((r : ereal), (p : ereal)) = (𝓝 (r, p)).map (λp:ℝ × ℝ, (p.1, p.2)) | ((open_embedding_coe.prod open_embedding_coe).map_nhds_eq (r, p)).symm | lemma | ereal.nhds_coe_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_to_real {a : ereal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥) :
tendsto ereal.to_real (𝓝 a) (𝓝 a.to_real) | begin
lift a to ℝ using and.intro ha h'a,
rw [nhds_coe, tendsto_map'_iff],
exact tendsto_id
end | lemma | ereal.tendsto_to_real | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"ereal.to_real",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_to_real : continuous_on ereal.to_real ({⊥, ⊤}ᶜ : set ereal) | λ a ha, continuous_at.continuous_within_at (tendsto_to_real
(by { simp [not_or_distrib] at ha, exact ha.2 }) (by { simp [not_or_distrib] at ha, exact ha.1 })) | lemma | ereal.continuous_on_to_real | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous_at.continuous_within_at",
"continuous_on",
"ereal",
"ereal.to_real",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot_top_homeomorph_real : ({⊥, ⊤}ᶜ : set ereal) ≃ₜ ℝ | { continuous_to_fun := continuous_on_iff_continuous_restrict.1 continuous_on_to_real,
continuous_inv_fun := continuous_coe_real_ereal.subtype_mk _,
.. ne_top_bot_equiv_real } | def | ereal.ne_bot_top_homeomorph_real | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal"
] | The set of finite `ereal` numbers is homeomorphic to `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_coe_ennreal : embedding (coe : ℝ≥0∞ → ereal) | ⟨⟨begin
refine le_antisymm _ _,
{ rw [@order_topology.topology_eq_generate_intervals ereal _,
← coinduced_le_iff_le_induced],
refine le_generate_from (assume s ha, _),
rcases ha with ⟨a, rfl | rfl⟩,
show is_open {b : ℝ≥0∞ | a < ↑b},
{ induction a using ereal.rec with x,
{ simp only [is_o... | lemma | ereal.embedding_coe_ennreal | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"coinduced_le_iff_le_induced",
"embedding",
"ereal",
"ereal.rec",
"imp_self",
"is_open",
"is_open_Iio",
"is_open_Ioi",
"is_open_empty",
"is_open_univ",
"le_generate_from",
"not_lt_bot",
"not_top_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe_ennreal {α : Type*} {f : filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} :
tendsto (λ a, (m a : ereal)) f (𝓝 ↑a) ↔ tendsto m f (𝓝 a) | embedding_coe_ennreal.tendsto_nhds_iff.symm | lemma | ereal.tendsto_coe_ennreal | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_coe_ennreal_ereal : continuous (coe : ℝ≥0∞ → ereal) | embedding_coe_ennreal.continuous | lemma | continuous_coe_ennreal_ereal | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe_ennreal_iff {f : α → ℝ≥0∞} :
continuous (λa, (f a : ereal)) ↔ continuous f | embedding_coe_ennreal.continuous_iff.symm | lemma | ereal.continuous_coe_ennreal_iff | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top : 𝓝 (⊤ : ereal) = ⨅ a ≠ ⊤, 𝓟 (Ioi a) | nhds_top_order.trans $ by simp [lt_top_iff_ne_top, Ioi] | lemma | ereal.nhds_top | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"lt_top_iff_ne_top",
"nhds_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top' : 𝓝 (⊤ : ereal) = ⨅ a : ℝ, 𝓟 (Ioi a) | begin
rw [nhds_top],
apply le_antisymm,
{ exact infi_mono' (λ x, ⟨x, by simp⟩) },
{ refine le_infi (λ r, le_infi (λ hr, _)),
induction r using ereal.rec,
{ exact (infi_le _ 0).trans (by simp) },
{ exact infi_le _ _ },
{ simpa using hr, } }
end | lemma | ereal.nhds_top' | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"ereal.rec",
"infi_le",
"infi_mono'",
"le_infi",
"nhds_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_top_iff {s : set ereal} :
s ∈ 𝓝 (⊤ : ereal) ↔ ∃ (y : ℝ), Ioi (y : ereal) ⊆ s | begin
rw [nhds_top', mem_infi_of_directed],
{ refl },
exact λ x y, ⟨max x y, by simp [le_refl], by simp [le_refl]⟩,
end | lemma | ereal.mem_nhds_top_iff | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_top_iff_real {α : Type*} {m : α → ereal} {f : filter α} :
tendsto m f (𝓝 ⊤) ↔ ∀ x : ℝ, ∀ᶠ a in f, ↑x < m a | by simp only [nhds_top', mem_Ioi, tendsto_infi, tendsto_principal] | lemma | ereal.tendsto_nhds_top_iff_real | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bot : 𝓝 (⊥ : ereal) = ⨅ a ≠ ⊥, 𝓟 (Iio a) | nhds_bot_order.trans $ by simp [bot_lt_iff_ne_bot] | lemma | ereal.nhds_bot | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"bot_lt_iff_ne_bot",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_bot' : 𝓝 (⊥ : ereal) = ⨅ a : ℝ, 𝓟 (Iio a) | begin
rw [nhds_bot],
apply le_antisymm,
{ exact infi_mono' (λ x, ⟨x, by simp⟩) },
{ refine le_infi (λ r, le_infi (λ hr, _)),
induction r using ereal.rec,
{ simpa using hr },
{ exact infi_le _ _ },
{ exact (infi_le _ 0).trans (by simp) } }
end | lemma | ereal.nhds_bot' | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"ereal.rec",
"infi_le",
"infi_mono'",
"le_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_bot_iff {s : set ereal} :
s ∈ 𝓝 (⊥ : ereal) ↔ ∃ (y : ℝ), Iio (y : ereal) ⊆ s | begin
rw [nhds_bot', mem_infi_of_directed],
{ refl },
exact λ x y, ⟨min x y, by simp [le_refl], by simp [le_refl]⟩,
end | lemma | ereal.mem_nhds_bot_iff | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_bot_iff_real {α : Type*} {m : α → ereal} {f : filter α} :
tendsto m f (𝓝 ⊥) ↔ ∀ x : ℝ, ∀ᶠ a in f, m a < x | by simp only [nhds_bot', mem_Iio, tendsto_infi, tendsto_principal] | lemma | ereal.tendsto_nhds_bot_iff_real | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_coe_coe (a b :ℝ) :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (a, b) | by simp only [continuous_at, nhds_coe_coe, ← coe_add, tendsto_map'_iff, (∘),
tendsto_coe, tendsto_add] | lemma | ereal.continuous_at_add_coe_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous_at",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_top_coe (a : ℝ) :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (⊤, a) | begin
simp only [continuous_at, tendsto_nhds_top_iff_real, top_add_coe, nhds_prod_eq],
assume r,
rw eventually_prod_iff,
refine ⟨λ z, ((r - (a - 1): ℝ) : ereal) < z, Ioi_mem_nhds (coe_lt_top _),
λ z, ((a - 1 : ℝ) : ereal) < z, Ioi_mem_nhds (by simp [-ereal.coe_sub]),
λ x hx y hy, _⟩,
dsimp... | lemma | ereal.continuous_at_add_top_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"Ioi_mem_nhds",
"continuous_at",
"ereal",
"ereal.coe_sub",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_coe_top (a : ℝ) :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (a, ⊤) | begin
change continuous_at ((λ (p : ereal × ereal), p.2 + p.1) ∘ prod.swap) (a, ⊤),
apply continuous_at.comp _ continuous_swap.continuous_at,
simp_rw add_comm,
exact continuous_at_add_top_coe a
end | lemma | ereal.continuous_at_add_coe_top | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous_at",
"continuous_at.comp",
"ereal",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_top_top :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (⊤, ⊤) | begin
simp only [continuous_at, tendsto_nhds_top_iff_real, top_add_top, nhds_prod_eq],
assume r,
rw eventually_prod_iff,
refine ⟨λ z, (r : ereal) < z, Ioi_mem_nhds (coe_lt_top _),
λ z, ((0 : ℝ) : ereal) < z, Ioi_mem_nhds (by simp [zero_lt_one]),
λ x hx y hy, _⟩,
dsimp,
convert add_lt_add... | lemma | ereal.continuous_at_add_top_top | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"Ioi_mem_nhds",
"continuous_at",
"ereal",
"nhds_prod_eq",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_bot_coe (a : ℝ) :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (⊥, a) | begin
simp only [continuous_at, tendsto_nhds_bot_iff_real, nhds_prod_eq, bot_add],
assume r,
rw eventually_prod_iff,
refine ⟨λ z, z < ((r - (a + 1): ℝ) : ereal), Iio_mem_nhds (bot_lt_coe _),
λ z, z < ((a + 1 : ℝ) : ereal), Iio_mem_nhds (by simp [-coe_add, zero_lt_one]),
λ x hx y hy, _⟩,
co... | lemma | ereal.continuous_at_add_bot_coe | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"Iio_mem_nhds",
"continuous_at",
"ereal",
"nhds_prod_eq",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_coe_bot (a : ℝ) :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (a, ⊥) | begin
change continuous_at ((λ (p : ereal × ereal), p.2 + p.1) ∘ prod.swap) (a, ⊥),
apply continuous_at.comp _ continuous_swap.continuous_at,
simp_rw add_comm,
exact continuous_at_add_bot_coe a
end | lemma | ereal.continuous_at_add_coe_bot | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous_at",
"continuous_at.comp",
"ereal",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add_bot_bot :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) (⊥, ⊥) | begin
simp only [continuous_at, tendsto_nhds_bot_iff_real, nhds_prod_eq, bot_add],
assume r,
rw eventually_prod_iff,
refine ⟨λ z, z < r, Iio_mem_nhds (bot_lt_coe _),
λ z, z < 0, Iio_mem_nhds (bot_lt_coe _),
λ x hx y hy, _⟩,
dsimp,
convert add_lt_add hx hy,
simp
end | lemma | ereal.continuous_at_add_bot_bot | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"Iio_mem_nhds",
"continuous_at",
"ereal",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_add {p : ereal × ereal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤) :
continuous_at (λ (p : ereal × ereal), p.1 + p.2) p | begin
rcases p with ⟨x, y⟩,
induction x using ereal.rec; induction y using ereal.rec,
{ exact continuous_at_add_bot_bot },
{ exact continuous_at_add_bot_coe _ },
{ simpa using h' },
{ exact continuous_at_add_coe_bot _ },
{ exact continuous_at_add_coe_coe _ _ },
{ exact continuous_at_add_coe_top _ },
{... | lemma | ereal.continuous_at_add | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous_at",
"ereal",
"ereal.rec"
] | The addition on `ereal` is continuous except where it doesn't make sense (i.e., at `(⊥, ⊤)`
and at `(⊤, ⊥)`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
neg_homeo : ereal ≃ₜ ereal | neg_order_iso.to_homeomorph | def | ereal.neg_homeo | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"ereal"
] | Negation on `ereal` as a homeomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_neg : continuous (λ (x : ereal), -x) | neg_homeo.continuous | lemma | ereal.continuous_neg | topology.instances | src/topology/instances/ereal.lean | [
"data.rat.encodable",
"data.real.ereal",
"topology.algebra.order.monotone_continuity",
"topology.instances.ennreal"
] | [
"continuous",
"ereal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq (x y : ℤ) : dist x y = |x - y| | rfl | theorem | int.dist_eq | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_cast_real (x y : ℤ) : dist (x : ℝ) y = dist x y | rfl | theorem | int.dist_cast_real | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pairwise_one_le_dist : pairwise (λ m n : ℤ, 1 ≤ dist m n) | begin
intros m n hne,
rw dist_eq, norm_cast, rwa [← zero_add (1 : ℤ), int.add_one_le_iff, abs_pos, sub_ne_zero]
end | lemma | int.pairwise_one_le_dist | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"abs_pos",
"int.add_one_le_iff",
"pairwise"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_embedding_coe_real : uniform_embedding (coe : ℤ → ℝ) | uniform_embedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | lemma | int.uniform_embedding_coe_real | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"uniform_embedding",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_coe_real : closed_embedding (coe : ℤ → ℝ) | closed_embedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist | lemma | int.closed_embedding_coe_real | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"closed_embedding",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_ball (x : ℤ) (r : ℝ) : coe ⁻¹' (ball (x : ℝ) r) = ball x r | rfl | theorem | int.preimage_ball | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closed_ball (x : ℤ) (r : ℝ) :
coe ⁻¹' (closed_ball (x : ℝ) r) = closed_ball x r | rfl | theorem | int.preimage_closed_ball | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_eq_Ioo (x : ℤ) (r : ℝ) : ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉ | by rw [← preimage_ball, real.ball_eq_Ioo, preimage_Ioo] | theorem | int.ball_eq_Ioo | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"real.ball_eq_Ioo"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_eq_Icc (x : ℤ) (r : ℝ) : closed_ball x r = Icc ⌈↑x - r⌉ ⌊↑x + r⌋ | by rw [← preimage_closed_ball, real.closed_ball_eq_Icc, preimage_Icc] | theorem | int.closed_ball_eq_Icc | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"real.closed_ball_eq_Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocompact_eq : cocompact ℤ = at_bot ⊔ at_top | by simp only [← comap_dist_right_at_top_eq_cocompact (0 : ℤ), dist_eq, sub_zero, cast_zero,
← cast_abs, ← @comap_comap _ _ _ _ abs, int.comap_coe_at_top, comap_abs_at_top] | lemma | int.cocompact_eq | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"comap_dist_right_at_top_eq_cocompact",
"int.comap_coe_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cofinite_eq : (cofinite : filter ℤ) = at_bot ⊔ at_top | by rw [← cocompact_eq_cofinite, cocompact_eq] | lemma | int.cofinite_eq | topology.instances | src/topology/instances/int.lean | [
"data.int.interval",
"topology.metric_space.basic",
"order.filter.archimedean"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Gδ_irrational : is_Gδ {x | irrational x} | (countable_range _).is_Gδ_compl | lemma | is_Gδ_irrational | topology.instances | src/topology/instances/irrational.lean | [
"data.real.irrational",
"data.rat.encodable",
"topology.metric_space.baire"
] | [
"irrational",
"is_Gδ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_irrational : dense {x : ℝ | irrational x} | begin
refine real.is_topological_basis_Ioo_rat.dense_iff.2 _,
simp only [mem_Union, mem_singleton_iff],
rintro _ ⟨a, b, hlt, rfl⟩ hne, rw inter_comm,
exact exists_irrational_btwn (rat.cast_lt.2 hlt)
end | lemma | dense_irrational | topology.instances | src/topology/instances/irrational.lean | [
"data.real.irrational",
"data.rat.encodable",
"topology.metric_space.baire"
] | [
"dense",
"exists_irrational_btwn",
"irrational"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_residual_irrational : ∀ᶠ x in residual ℝ, irrational x | eventually_residual.2 ⟨_, is_Gδ_irrational, dense_irrational, λ _, id⟩ | lemma | eventually_residual_irrational | topology.instances | src/topology/instances/irrational.lean | [
"data.real.irrational",
"data.rat.encodable",
"topology.metric_space.baire"
] | [
"dense_irrational",
"irrational",
"is_Gδ_irrational",
"residual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_forall_le_dist_cast_div (hx : irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n) | begin
have A : is_closed (range (λ m, n⁻¹ * m : ℤ → ℝ)),
from ((is_closed_map_smul₀ (n⁻¹ : ℝ)).comp
int.closed_embedding_coe_real.is_closed_map).closed_range,
have B : x ∉ range (λ m, n⁻¹ * m : ℤ → ℝ),
{ rintro ⟨m, rfl⟩, simpa using hx },
rcases metric.mem_nhds_iff.1 (A.is_open_compl.mem_nhds B) with ... | lemma | irrational.eventually_forall_le_dist_cast_div | topology.instances | src/topology/instances/irrational.lean | [
"data.real.irrational",
"data.rat.encodable",
"topology.metric_space.baire"
] | [
"dist_comm",
"div_eq_inv_mul",
"ge_mem_nhds",
"irrational",
"is_closed",
"is_closed_map_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_forall_le_dist_cast_div_of_denom_le (hx : irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ (k ≤ n) (m : ℤ), ε ≤ dist x (m / k) | (finite_le_nat n).eventually_all.2 $ λ k hk, hx.eventually_forall_le_dist_cast_div k | lemma | irrational.eventually_forall_le_dist_cast_div_of_denom_le | topology.instances | src/topology/instances/irrational.lean | [
"data.real.irrational",
"data.rat.encodable",
"topology.metric_space.baire"
] | [
"irrational"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_forall_le_dist_cast_rat_of_denom_le (hx : irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ r : ℚ, r.denom ≤ n → ε ≤ dist x r | (hx.eventually_forall_le_dist_cast_div_of_denom_le n).mono $ λ ε H r hr,
by simpa only [rat.cast_def] using H r.denom hr r.num | lemma | irrational.eventually_forall_le_dist_cast_rat_of_denom_le | topology.instances | src/topology/instances/irrational.lean | [
"data.real.irrational",
"data.rat.encodable",
"topology.metric_space.baire"
] | [
"irrational",
"rat.cast_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_matrix [topological_space α] {f : α → matrix m n R}
(h : ∀ i j, continuous (λ a, f a i j)) : continuous f | continuous_pi $ λ _, continuous_pi $ λ j, h _ _ | lemma | continuous_matrix | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_pi",
"matrix",
"topological_space"
] | To show a function into matrices is continuous it suffices to show the coefficients of the
resulting matrix are continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.matrix_elem {A : X → matrix m n R} (hA : continuous A) (i : m) (j : n) :
continuous (λ x, A x i j) | (continuous_apply_apply i j).comp hA | lemma | continuous.matrix_elem | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply_apply",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_map [topological_space S] {A : X → matrix m n S} {f : S → R}
(hA : continuous A) (hf : continuous f) :
continuous (λ x, (A x).map f) | continuous_matrix $ λ i j, hf.comp $ hA.matrix_elem _ _ | lemma | continuous.matrix_map | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_matrix",
"matrix",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_transpose {A : X → matrix m n R} (hA : continuous A) :
continuous (λ x, (A x)ᵀ) | continuous_matrix $ λ i j, hA.matrix_elem j i | lemma | continuous.matrix_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_matrix",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_conj_transpose [has_star R] [has_continuous_star R] {A : X → matrix m n R}
(hA : continuous A) : continuous (λ x, (A x)ᴴ) | hA.matrix_transpose.matrix_map continuous_star | lemma | continuous.matrix_conj_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"has_continuous_star",
"has_star",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_col {A : X → n → R} (hA : continuous A) : continuous (λ x, col (A x)) | continuous_matrix $ λ i j, (continuous_apply _).comp hA | lemma | continuous.matrix_col | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_row {A : X → n → R} (hA : continuous A) : continuous (λ x, row (A x)) | continuous_matrix $ λ i j, (continuous_apply _).comp hA | lemma | continuous.matrix_row | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_diagonal [has_zero R] [decidable_eq n] {A : X → n → R} (hA : continuous A) :
continuous (λ x, diagonal (A x)) | continuous_matrix $ λ i j, ((continuous_apply i).comp hA).if_const _ continuous_zero | lemma | continuous.matrix_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_dot_product [fintype n] [has_mul R] [add_comm_monoid R]
[has_continuous_add R] [has_continuous_mul R]
{A : X → n → R} {B : X → n → R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, dot_product (A x) (B x)) | continuous_finset_sum _ $ λ i _, ((continuous_apply i).comp hA).mul ((continuous_apply i).comp hB) | lemma | continuous.matrix_dot_product | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"add_comm_monoid",
"continuous",
"continuous_apply",
"fintype",
"has_continuous_add",
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_mul [fintype n] [has_mul R] [add_comm_monoid R] [has_continuous_add R]
[has_continuous_mul R]
{A : X → matrix m n R} {B : X → matrix n p R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, (A x).mul (B x)) | continuous_matrix $ λ i j, continuous_finset_sum _ $ λ k _,
(hA.matrix_elem _ _).mul (hB.matrix_elem _ _) | lemma | continuous.matrix_mul | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"add_comm_monoid",
"continuous",
"continuous_matrix",
"fintype",
"has_continuous_add",
"has_continuous_mul",
"matrix"
] | For square matrices the usual `continuous_mul` can be used. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.matrix_vec_mul_vec [has_mul R] [has_continuous_mul R]
{A : X → m → R} {B : X → n → R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, vec_mul_vec (A x) (B x)) | continuous_matrix $ λ i j, ((continuous_apply _).comp hA).mul ((continuous_apply _).comp hB) | lemma | continuous.matrix_vec_mul_vec | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_matrix",
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_mul_vec [non_unital_non_assoc_semiring R] [has_continuous_add R]
[has_continuous_mul R] [fintype n]
{A : X → matrix m n R} {B : X → n → R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, (A x).mul_vec (B x)) | continuous_pi $ λ i, ((continuous_apply i).comp hA).matrix_dot_product hB | lemma | continuous.matrix_mul_vec | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_pi",
"fintype",
"has_continuous_add",
"has_continuous_mul",
"matrix",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_vec_mul [non_unital_non_assoc_semiring R] [has_continuous_add R]
[has_continuous_mul R] [fintype m]
{A : X → m → R} {B : X → matrix m n R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, vec_mul (A x) (B x)) | continuous_pi $ λ i, hA.matrix_dot_product $ continuous_pi $ λ j, hB.matrix_elem _ _ | lemma | continuous.matrix_vec_mul | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_pi",
"fintype",
"has_continuous_add",
"has_continuous_mul",
"matrix",
"non_unital_non_assoc_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_submatrix
{A : X → matrix l n R} (hA : continuous A) (e₁ : m → l) (e₂ : p → n) :
continuous (λ x, (A x).submatrix e₁ e₂) | continuous_matrix $ λ i j, hA.matrix_elem _ _ | lemma | continuous.matrix_submatrix | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_matrix",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_reindex {A : X → matrix l n R}
(hA : continuous A) (e₁ : l ≃ m) (e₂ : n ≃ p) :
continuous (λ x, reindex e₁ e₂ (A x)) | hA.matrix_submatrix _ _ | lemma | continuous.matrix_reindex | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"matrix",
"reindex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_diag {A : X → matrix n n R} (hA : continuous A) :
continuous (λ x, matrix.diag (A x)) | continuous_pi $ λ _, hA.matrix_elem _ _ | lemma | continuous.matrix_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_pi",
"matrix",
"matrix.diag"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_matrix_diag : continuous (matrix.diag : matrix n n R → n → R) | show continuous (λ x : matrix n n R, matrix.diag x), from continuous_id.matrix_diag | lemma | continuous_matrix_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"matrix",
"matrix.diag"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_trace [fintype n] [add_comm_monoid R] [has_continuous_add R]
{A : X → matrix n n R} (hA : continuous A) :
continuous (λ x, trace (A x)) | continuous_finset_sum _ $ λ i hi, hA.matrix_elem _ _ | lemma | continuous.matrix_trace | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"add_comm_monoid",
"continuous",
"fintype",
"has_continuous_add",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_det [fintype n] [decidable_eq n] [comm_ring R] [topological_ring R]
{A : X → matrix n n R} (hA : continuous A) :
continuous (λ x, (A x).det) | begin
simp_rw matrix.det_apply,
refine continuous_finset_sum _ (λ l _, continuous.const_smul _ _),
refine continuous_finset_prod _ (λ l _, hA.matrix_elem _ _),
end | lemma | continuous.matrix_det | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"comm_ring",
"continuous",
"continuous.const_smul",
"continuous_finset_prod",
"fintype",
"matrix",
"matrix.det_apply",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_update_column [decidable_eq n] (i : n)
{A : X → matrix m n R} {B : X → m → R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, (A x).update_column i (B x)) | continuous_matrix $ λ j k, (continuous_apply k).comp $
((continuous_apply _).comp hA).update i ((continuous_apply _).comp hB) | lemma | continuous.matrix_update_column | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_matrix",
"matrix",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_update_row [decidable_eq m] (i : m)
{A : X → matrix m n R} {B : X → n → R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, (A x).update_row i (B x)) | hA.update i hB | lemma | continuous.matrix_update_row | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_cramer [fintype n] [decidable_eq n] [comm_ring R] [topological_ring R]
{A : X → matrix n n R} {B : X → n → R} (hA : continuous A) (hB : continuous B) :
continuous (λ x, (A x).cramer (B x)) | continuous_pi $ λ i, (hA.matrix_update_column _ hB).matrix_det | lemma | continuous.matrix_cramer | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"comm_ring",
"continuous",
"continuous_pi",
"fintype",
"matrix",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_adjugate [fintype n] [decidable_eq n] [comm_ring R] [topological_ring R]
{A : X → matrix n n R} (hA : continuous A) :
continuous (λ x, (A x).adjugate) | continuous_matrix $ λ j k, (hA.matrix_transpose.matrix_update_column k continuous_const).matrix_det | lemma | continuous.matrix_adjugate | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"comm_ring",
"continuous",
"continuous_const",
"continuous_matrix",
"fintype",
"matrix",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_matrix_inv [fintype n] [decidable_eq n] [comm_ring R] [topological_ring R]
(A : matrix n n R) (h : continuous_at ring.inverse A.det) :
continuous_at has_inv.inv A | (h.comp continuous_id.matrix_det.continuous_at).smul continuous_id.matrix_adjugate.continuous_at | lemma | continuous_at_matrix_inv | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"comm_ring",
"continuous_at",
"fintype",
"matrix",
"ring.inverse",
"topological_ring"
] | When `ring.inverse` is continuous at the determinant (such as in a `normed_ring`, or a
`topological_field`), so is `matrix.has_inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.matrix_from_blocks
{A : X → matrix n l R} {B : X → matrix n m R} {C : X → matrix p l R} {D : X → matrix p m R}
(hA : continuous A) (hB : continuous B) (hC : continuous C) (hD : continuous D) :
continuous (λ x, matrix.from_blocks (A x) (B x) (C x) (D x)) | continuous_matrix $ λ i j,
by cases i; cases j; refine continuous.matrix_elem _ i j; assumption | lemma | continuous.matrix_from_blocks | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous.matrix_elem",
"continuous_matrix",
"matrix",
"matrix.from_blocks"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_block_diagonal [has_zero R] [decidable_eq p] {A : X → p → matrix m n R}
(hA : continuous A) :
continuous (λ x, block_diagonal (A x)) | continuous_matrix $ λ ⟨i₁, i₂⟩ ⟨j₁, j₂⟩,
(((continuous_apply i₂).comp hA).matrix_elem i₁ j₁).if_const _ continuous_zero | lemma | continuous.matrix_block_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_matrix",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_block_diag {A : X → matrix (m × p) (n × p) R} (hA : continuous A) :
continuous (λ x, block_diag (A x)) | continuous_pi $ λ i, continuous_matrix $ λ j k, hA.matrix_elem _ _ | lemma | continuous.matrix_block_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_matrix",
"continuous_pi",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_block_diagonal' [has_zero R] [decidable_eq l]
{A : X → Π i, matrix (m' i) (n' i) R} (hA : continuous A) :
continuous (λ x, block_diagonal' (A x)) | continuous_matrix $ λ ⟨i₁, i₂⟩ ⟨j₁, j₂⟩, begin
dsimp only [block_diagonal'_apply'],
split_ifs,
{ subst h,
exact ((continuous_apply i₁).comp hA).matrix_elem i₂ j₂ },
{ exact continuous_const },
end | lemma | continuous.matrix_block_diagonal' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_apply",
"continuous_const",
"continuous_matrix",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.matrix_block_diag' {A : X → matrix (Σ i, m' i) (Σ i, n' i) R} (hA : continuous A) :
continuous (λ x, block_diag' (A x)) | continuous_pi $ λ i, continuous_matrix $ λ j k, hA.matrix_elem _ _ | lemma | continuous.matrix_block_diag' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous",
"continuous_matrix",
"continuous_pi",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_transpose {f : X → matrix m n R} {a : matrix m n R} (hf : has_sum f a) :
has_sum (λ x, (f x)ᵀ) aᵀ | (hf.map (matrix.transpose_add_equiv m n R) continuous_id.matrix_transpose : _) | lemma | has_sum.matrix_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"has_sum",
"matrix",
"matrix.transpose_add_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_transpose {f : X → matrix m n R} (hf : summable f) :
summable (λ x, (f x)ᵀ) | hf.has_sum.matrix_transpose.summable | lemma | summable.matrix_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_matrix_transpose {f : X → matrix m n R} :
summable (λ x, (f x)ᵀ) ↔ summable f | (summable.map_iff_of_equiv (matrix.transpose_add_equiv m n R)
(@continuous_id (matrix m n R) _).matrix_transpose (continuous_id.matrix_transpose) : _) | lemma | summable_matrix_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous_id",
"matrix",
"matrix.transpose_add_equiv",
"summable",
"summable.map_iff_of_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.transpose_tsum [t2_space R] {f : X → matrix m n R} : (∑' x, f x)ᵀ = ∑' x, (f x)ᵀ | begin
by_cases hf : summable f,
{ exact hf.has_sum.matrix_transpose.tsum_eq.symm },
{ have hft := summable_matrix_transpose.not.mpr hf,
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, transpose_zero] },
end | lemma | matrix.transpose_tsum | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_conj_transpose [star_add_monoid R] [has_continuous_star R]
{f : X → matrix m n R} {a : matrix m n R} (hf : has_sum f a) :
has_sum (λ x, (f x)ᴴ) aᴴ | (hf.map (matrix.conj_transpose_add_equiv m n R) continuous_id.matrix_conj_transpose : _) | lemma | has_sum.matrix_conj_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"has_continuous_star",
"has_sum",
"matrix",
"matrix.conj_transpose_add_equiv",
"star_add_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_conj_transpose [star_add_monoid R] [has_continuous_star R]
{f : X → matrix m n R} (hf : summable f) :
summable (λ x, (f x)ᴴ) | hf.has_sum.matrix_conj_transpose.summable | lemma | summable.matrix_conj_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"has_continuous_star",
"matrix",
"star_add_monoid",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_matrix_conj_transpose [star_add_monoid R] [has_continuous_star R]
{f : X → matrix m n R} :
summable (λ x, (f x)ᴴ) ↔ summable f | (summable.map_iff_of_equiv (matrix.conj_transpose_add_equiv m n R)
(@continuous_id (matrix m n R) _).matrix_conj_transpose (continuous_id.matrix_conj_transpose) : _) | lemma | summable_matrix_conj_transpose | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous_id",
"has_continuous_star",
"matrix",
"matrix.conj_transpose_add_equiv",
"star_add_monoid",
"summable",
"summable.map_iff_of_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.conj_transpose_tsum [star_add_monoid R] [has_continuous_star R] [t2_space R]
{f : X → matrix m n R} : (∑' x, f x)ᴴ = ∑' x, (f x)ᴴ | begin
by_cases hf : summable f,
{ exact hf.has_sum.matrix_conj_transpose.tsum_eq.symm },
{ have hft := summable_matrix_conj_transpose.not.mpr hf,
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft, conj_transpose_zero] },
end | lemma | matrix.conj_transpose_tsum | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"has_continuous_star",
"matrix",
"star_add_monoid",
"summable",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_diagonal [decidable_eq n] {f : X → n → R} {a : n → R} (hf : has_sum f a) :
has_sum (λ x, diagonal (f x)) (diagonal a) | (hf.map (diagonal_add_monoid_hom n R) $ continuous.matrix_diagonal $ by exact continuous_id : _) | lemma | has_sum.matrix_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_diagonal",
"continuous_id",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_diagonal [decidable_eq n] {f : X → n → R} (hf : summable f) :
summable (λ x, diagonal (f x)) | hf.has_sum.matrix_diagonal.summable | lemma | summable.matrix_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_matrix_diagonal [decidable_eq n] {f : X → n → R} :
summable (λ x, diagonal (f x)) ↔ summable f | (summable.map_iff_of_left_inverse
(@matrix.diagonal_add_monoid_hom n R _ _) (matrix.diag_add_monoid_hom n R)
(by exact continuous.matrix_diagonal continuous_id)
continuous_matrix_diag
(λ A, diag_diagonal A) : _) | lemma | summable_matrix_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_diagonal",
"continuous_id",
"continuous_matrix_diag",
"matrix.diag_add_monoid_hom",
"matrix.diagonal_add_monoid_hom",
"summable",
"summable.map_iff_of_left_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.diagonal_tsum [decidable_eq n] [t2_space R] {f : X → n → R} :
diagonal (∑' x, f x) = ∑' x, diagonal (f x) | begin
by_cases hf : summable f,
{ exact hf.has_sum.matrix_diagonal.tsum_eq.symm },
{ have hft := summable_matrix_diagonal.not.mpr hf,
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft],
exact diagonal_zero },
end | lemma | matrix.diagonal_tsum | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"summable",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_diag {f : X → matrix n n R} {a : matrix n n R} (hf : has_sum f a) :
has_sum (λ x, diag (f x)) (diag a) | (hf.map (diag_add_monoid_hom n R) continuous_matrix_diag : _) | lemma | has_sum.matrix_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous_matrix_diag",
"has_sum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_diag {f : X → matrix n n R} (hf : summable f) : summable (λ x, diag (f x)) | hf.has_sum.matrix_diag.summable | lemma | summable.matrix_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_block_diagonal [decidable_eq p]
{f : X → p → matrix m n R} {a : p → matrix m n R} (hf : has_sum f a) :
has_sum (λ x, block_diagonal (f x)) (block_diagonal a) | (hf.map (block_diagonal_add_monoid_hom m n p R) $
continuous.matrix_block_diagonal $ by exact continuous_id : _) | lemma | has_sum.matrix_block_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_block_diagonal",
"continuous_id",
"has_sum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_block_diagonal [decidable_eq p] {f : X → p → matrix m n R} (hf : summable f) :
summable (λ x, block_diagonal (f x)) | hf.has_sum.matrix_block_diagonal.summable | lemma | summable.matrix_block_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_matrix_block_diagonal [decidable_eq p] {f : X → p → matrix m n R} :
summable (λ x, block_diagonal (f x)) ↔ summable f | (summable.map_iff_of_left_inverse
(matrix.block_diagonal_add_monoid_hom m n p R) (matrix.block_diag_add_monoid_hom m n p R)
(by exact continuous.matrix_block_diagonal continuous_id)
(by exact continuous.matrix_block_diag continuous_id)
(λ A, block_diag_block_diagonal A) : _) | lemma | summable_matrix_block_diagonal | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_block_diag",
"continuous.matrix_block_diagonal",
"continuous_id",
"matrix",
"matrix.block_diag_add_monoid_hom",
"matrix.block_diagonal_add_monoid_hom",
"summable",
"summable.map_iff_of_left_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.block_diagonal_tsum [decidable_eq p] [t2_space R] {f : X → p → matrix m n R} :
block_diagonal (∑' x, f x) = ∑' x, block_diagonal (f x) | begin
by_cases hf : summable f,
{ exact hf.has_sum.matrix_block_diagonal.tsum_eq.symm },
{ have hft := summable_matrix_block_diagonal.not.mpr hf,
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft],
exact block_diagonal_zero },
end | lemma | matrix.block_diagonal_tsum | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_block_diag {f : X → matrix (m × p) (n × p) R}
{a : matrix (m × p) (n × p) R} (hf : has_sum f a) :
has_sum (λ x, block_diag (f x)) (block_diag a) | (hf.map (block_diag_add_monoid_hom m n p R) $ continuous.matrix_block_diag continuous_id : _) | lemma | has_sum.matrix_block_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_block_diag",
"continuous_id",
"has_sum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_block_diag {f : X → matrix (m × p) (n × p) R} (hf : summable f) :
summable (λ x, block_diag (f x)) | hf.has_sum.matrix_block_diag.summable | lemma | summable.matrix_block_diag | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_block_diagonal' [decidable_eq l]
{f : X → Π i, matrix (m' i) (n' i) R} {a : Π i, matrix (m' i) (n' i) R} (hf : has_sum f a) :
has_sum (λ x, block_diagonal' (f x)) (block_diagonal' a) | (hf.map (block_diagonal'_add_monoid_hom m' n' R) $
continuous.matrix_block_diagonal' $ by exact continuous_id : _) | lemma | has_sum.matrix_block_diagonal' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_block_diagonal'",
"continuous_id",
"has_sum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_block_diagonal' [decidable_eq l]
{f : X → Π i, matrix (m' i) (n' i) R} (hf : summable f) :
summable (λ x, block_diagonal' (f x)) | hf.has_sum.matrix_block_diagonal'.summable | lemma | summable.matrix_block_diagonal' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_matrix_block_diagonal' [decidable_eq l] {f : X → Π i, matrix (m' i) (n' i) R} :
summable (λ x, block_diagonal' (f x)) ↔ summable f | (summable.map_iff_of_left_inverse
(matrix.block_diagonal'_add_monoid_hom m' n' R) (matrix.block_diag'_add_monoid_hom m' n' R)
(by exact continuous.matrix_block_diagonal' continuous_id)
(by exact continuous.matrix_block_diag' continuous_id)
(λ A, block_diag'_block_diagonal' A) : _) | lemma | summable_matrix_block_diagonal' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_block_diag'",
"continuous.matrix_block_diagonal'",
"continuous_id",
"matrix",
"matrix.block_diag'_add_monoid_hom",
"matrix.block_diagonal'_add_monoid_hom",
"summable",
"summable.map_iff_of_left_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
matrix.block_diagonal'_tsum [decidable_eq l] [t2_space R]
{f : X → Π i, matrix (m' i) (n' i) R} :
block_diagonal' (∑' x, f x) = ∑' x, block_diagonal' (f x) | begin
by_cases hf : summable f,
{ exact hf.has_sum.matrix_block_diagonal'.tsum_eq.symm },
{ have hft := summable_matrix_block_diagonal'.not.mpr hf,
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hft],
exact block_diagonal'_zero },
end | lemma | matrix.block_diagonal'_tsum | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable",
"t2_space",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.matrix_block_diag' {f : X → matrix (Σ i, m' i) (Σ i, n' i) R}
{a : matrix (Σ i, m' i) (Σ i, n' i) R} (hf : has_sum f a) :
has_sum (λ x, block_diag' (f x)) (block_diag' a) | (hf.map (block_diag'_add_monoid_hom m' n' R) $ continuous.matrix_block_diag' continuous_id : _) | lemma | has_sum.matrix_block_diag' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"continuous.matrix_block_diag'",
"continuous_id",
"has_sum",
"matrix"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.matrix_block_diag' {f : X → matrix (Σ i, m' i) (Σ i, n' i) R} (hf : summable f) :
summable (λ x, block_diag' (f x)) | hf.has_sum.matrix_block_diag'.summable | lemma | summable.matrix_block_diag' | topology.instances | src/topology/instances/matrix.lean | [
"topology.algebra.infinite_sum.basic",
"topology.algebra.ring.basic",
"topology.algebra.star",
"linear_algebra.matrix.nonsingular_inverse",
"linear_algebra.matrix.trace"
] | [
"matrix",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_eq (x y : ℕ) : dist x y = |x - y| | rfl | theorem | nat.dist_eq | topology.instances | src/topology/instances/nat.lean | [
"topology.instances.int"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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